# Normal not implies normal-extensible automorphism-invariant in finite

This article gives the statement and possibly, proof, of a non-implication relation between two subgroup properties, when the big group is a finite group. That is, it states that in a finite group, every subgroup satisfying the first subgroup property (i.e., normal subgroup) neednotsatisfy the second subgroup property (i.e., normal-extensible automorphism-invariant subgroup)

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## Contents

## Statement

### Statement with symbols

It is possible to have a finite group , a normal subgroup of , and a normal-extensible automorphism of such that .

## Related facts

### Weaker facts

- Normal-extensible not implies normal
- Normal-extensible not implies extensible
- Normal-extensible not implies inner

### Applications

- Normal not implies semi-strongly potentially relatively characteristic
- Potentially characteristic not implies semi-strongly potentially relatively characteristic
- Normal not implies semi-strongly potentially characteristic, normal not implies strongly potentially characteristic
- Potentially characteristic not implies semi-strongly potentially characteristic, potentially characteristic not implies strongly potentially characteristic

## Facts used

- Every automorphism is center-fixing and inner automorphism group is maximal in automorphism group implies every automorphism is normal-extensible
- Automorphism group of direct power of simple non-abelian group equals wreath product of automorphism group and symmetric group

## Proof

### Example of the dihedral group

`Further information: dihedral group:D8, subgroup structure of dihedral group:D8`

Let be the dihedral group of order eight. Then, every automorphism of fixes every element of the center of , and also, the inner automorphism group of is maximal in the automorphism group of . Thus, by fact (1), every automorphism of is normal-extensible.

However, there is an automorphism of that interchanges the two normal Klein four-subgroups. Thus, these two normal subgroups are not invariant under this automorphism, and hence, we have an automorphism of that is normal-extensible but not normal.

Equivalently, the Klein four-subgroups are examples of normal subgroups that are not normal-extensible automorphism-invariant.

### Example involving a simple complete group

Let be a simple complete group. In other words, is a centerless simple group such that every automorphism of is inner. Let . By fact (2), the automorphism group of is the wreath product of with the symmetric group of degree two, which has , the inner automorphism group, as a subgroup of index two. Moreover, is centerless. Thus, by fact (1), we get that every automorphism of is normal-extensible.

However, the *coordinate exchange* automorphism of , that interchanges the two copies of , is not a normal automorphism because it interchanges these two normal subgroups. Thus, we have an example of a normal-extensible automorphism that is not normal.

Equivalently, either of the direct factors is an example of a normal subgroup that is not normal-extensible automorphism-invariant.